Ribbon graphs and meromorphic functions

TL;DR

This paper explores the relationship between ribbon graphs on Riemann surfaces and meromorphic functions, constructing surfaces with prescribed graph properties and analyzing self-intersection bounds.

Contribution

It introduces a method to construct Riemann surfaces with meromorphic functions from given ribbon graphs and investigates their topological and combinatorial properties.

Findings

Constructed Riemann surfaces with prescribed ribbon graph genus.

Showed the number of self-intersections cannot be controlled solely by surface genus.

Established lower bounds for self-intersections in meromorphic function images.

Abstract

Let Y be a compact Riemann surface, phi:Y -> CP^1 a meromorphic function, and Gamma in Y a ribbon graph avoiding the critical points of phi. Then phi(Gamma) is an immersed graph in CP^1. Conversely, given an immersion im:Theta to bCP^1 of an abstract multigraph Theta without vertices of valence 1 or 2, we describe a construction of a compact Riemann surface Y and a meromorphic function phi_{im}:Y in CP^1 such that phi_{im}(Gamma)=im(Theta). We investigate the relation between the topology of Y and the combinatorics of Gamma. In particular, for a surface of genus g we construct spanning ribbon graphs whose underlying abstract graphs have arbitrary prescribed graph genus g' smaller or equal g, including the planar case. As a consequence, the number of self-intersections of \phi(Gamma) cannot, in general, be controlled solely by the genus of Y. We establish general lower bounds for the number of self-intersections and formulate several open problems, with emphasis on planar ribbon graphs.